Elimination Math Problem: Solving x and y Equations

[nirvana1990]

Homework Statement

If x=2at/(1+t^2) and y=b(1-t^2)/(1+t^2), show that x^2/a^2+y^2/b^2=1

The Attempt at a Solution

I've tried squaring both equations: xt^2=4a^2t^2/(1+2t^2+t^4)

y^2=b^2(1-2t^2t^4)/(1+2t^2+t^4)

Now I've tried adding x^2 and y^2: 4a^2t^2+b^2(1-2t^2+t^4)/(1+2t^2+t^4)

Am I able to cancel any of this down now?

 

[Dick]

Try solving the y equation for t and substituting that t into the x equation.

 

[Kurdt]

I think you'll find it easier if you add x²/a² to y²/b².

Its made fairly easy since the denominators are the same and thus its just a manipulation of the numerator.

 

[EnumaElish]

First simplify each term x^2/a^2 and y^2/b^2.

Then expand the numerator of y^2/b^2. Add it to x^2/a^2, which has the identical denominator.

Can you get the rest?

 

[nirvana1990]

Oh yes thanks! x^2/a^2+y^2/b^2=4t^2+1-2t^2+t^4/1+2t^2+t^4
So the left hand side cancels to give 1.
Thanks again!